Rotation in coordinate geometry involves turning a shape about a fixed point, known as the center of rotation. In the context of the IGCSE syllabus, we focus on rotations around the origin $(0,0)$. This section covers $90°, 180°$, and $270°$ rotations, both clockwise (CW) and anti-clockwise (ACW), providing a formulaic approach to understand and apply these transformations.

7.1.2 Rotation

Rotation transformations are categorised based on the angle of rotation and the direction (CW or ACW). The origin acts as the pivot point for these rotations.

Image courtesy of Mashup Math

Rotation of 90°

• Clockwise (CW): A 90° CW rotation swaps the coordinates and changes the sign of the new x-coordinate.

• Anti-clockwise (ACW): A 90° ACW rotation swaps the coordinates and changes the sign of the new y-coordinate.

Rotation of 180°

• A 180° rotation (both CW and ACW) inverts both coordinates.

Rotation of 270°

• Clockwise (CW): Equivalent to a 90° ACW rotation from the original position.

• Anti-clockwise (ACW): Equivalent to a 90° CW rotation from the original position.

Worked Examples

Example 1: Rotation of 90° CW

Consider a point $P(2, 3)$ rotated 90° CW about the origin.

• Original Coordinates: $P(2, 3)$
• Apply 90° CW Rotation: $P'(3, -2)$

Example 2: Rotation of 90° ACW

Consider a point $P(2, 3)$ rotated 90° ACW about the origin.

• Original Coordinates: $P(2, 3)$
• Apply 90° ACW Rotation: $P'(-3, 2)$

Example 3: Rotation of 180°

Consider a point $P(2, 3)$ rotated 180° (either CW or ACW).

• Original Coordinates: $P(2, 3)$
• Apply 180° Rotation: $P'(-2, -3)$

Example 4: Rotation of 270° CW

Consider a point $P(2, 3)$ rotated 270° CW (equivalent to 90° ACW from the original position).

• Original Coordinates: $P(2, 3)$
• Apply 270° CW Rotation: $P'(-3, 2)$

Example 5: Rotation of 270° ACW

Consider a point $P(2, 3)$ rotated 270° ACW (equivalent to 90° CW from the original position).

• Original Coordinates: $P(2, 3)$
• Apply 270° ACW Rotation: $P'(3, -2)$

Practical Application

Problem: Rotate Triangle ABC

Given a triangle with vertices $A(1, 2)$, $B(3, 2)$, and $C(2, 4)$, rotate it 180° about the origin.

• Vertex A: $A(1, 2) \to A'(-1, -2)$
• Vertex B: $B(3, 2) \to B'(-3, -2)$
• Vertex C: $C(2, 4) \to C'(-2, -4)$

Problem: Rotate Square PQRS

A square has vertices $P(1, 1)$, $Q(1, -1)$, $R(-1, -1)$, and $S(-1, 1)$. Rotate it 90° CW.

• Vertex P: $P(1, 1) \to P'(1, -1)$
• Vertex Q: $Q(1, -1) \to Q'(-1, -1)$
• Vertex R: $R(-1, -1) \to R'(-1, 1)$
• Vertex S: $S(-1, 1) \to S'(1, 1)$